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<p>We consider some problems related to 2-orthogonal polynomials (2-OP). The solutions of these problems can be regarded as a supplement to the work [K. Douak and P. Maroni, “On a new class of 2-orthogonal polynomials, I: The recurrence relations and some properties,” <i>Integral Transforms and Special Functions</i> (2020)]. We revise the conditions imposed on the parameters of two original recurrence relations (the first of these conditions is for the 2-OP, while the second condition is for their normalized derivatives) and guaranteeing the “classical” nature of the 2-OP in Hahn’s sense. We constructively prove that these recurrence relations do not cover all possible “classical” 2-OPs. An example of “classical” 2-OPs generated by the generating function constructed with the help of a Bessel function of the first kind of order zero is presented. These OPs are unique because their properties are similar to the properties of the classical OPs. In particular, this concerns the fact that their zeros and their location are real. The analysis of the available literature and our own numerous numerical-analytic experiments reveals the absence of other examples of the “classical” 2-OPs with similar properties.</p>

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Supplement to the Article by Douak and Maroni (2020) on a New Class of 2-Orthogonal Polynomials

  • 期刊论文

  • 发表日期：

    2025年8月1日

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  • Volodymyr Makarov

摘要

We consider some problems related to 2-orthogonal polynomials (2-OP). The solutions of these problems can be regarded as a supplement to the work [K. Douak and P. Maroni, “On a new class of 2-orthogonal polynomials, I: The recurrence relations and some properties,” Integral Transforms and Special Functions (2020)]. We revise the conditions imposed on the parameters of two original recurrence relations (the first of these conditions is for the 2-OP, while the second condition is for their normalized derivatives) and guaranteeing the “classical” nature of the 2-OP in Hahn’s sense. We constructively prove that these recurrence relations do not cover all possible “classical” 2-OPs. An example of “classical” 2-OPs generated by the generating function constructed with the help of a Bessel function of the first kind of order zero is presented. These OPs are unique because their properties are similar to the properties of the classical OPs. In particular, this concerns the fact that their zeros and their location are real. The analysis of the available literature and our own numerous numerical-analytic experiments reveals the absence of other examples of the “classical” 2-OPs with similar properties.